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14
A deterministic subexponential algorithm for solving parity games
 SODA
, 2006
"... The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms ..."
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Cited by 80 (3 self)
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The existence of polynomial time algorithms for the solution of parity games is a major open problem. The fastest known algorithms for the problem are randomized algorithms that run in subexponential time. These algorithms are all ultimately based on the randomized subexponential simplex algorithms of Kalai and of Matousek, Sharir and Welzl. Randomness seems to play an essential role in these algorithms. We use a completely different, and elementary, approach to obtain a deterministic subexponential algorithm for the solution of parity games. The new algorithm, like the existing randomized subexponential algorithms, uses only polynomial space, and it is almost as fast as the randomized subexponential algorithms mentioned above.
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 22 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
Unique sink orientations of grids
 Proc. 11th Conference on Integer Programming and Combinatorial Optimization (IPCO
, 2005
"... We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of su ..."
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Cited by 11 (4 self)
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We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the HoltKlee condition known to hold for polytope digraphs, and we give the first expected lineartime algorithms for solving PGLCP with a fixed number of blocks.
Diameter of Polyhedra: Limits of Abstraction
, 2009
"... We investigate the diameter of a natural abstraction of the 1skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that ..."
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Cited by 9 (2 self)
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We investigate the diameter of a natural abstraction of the 1skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear lower bound.
Randomized Subexponential Algorithms for Infinite Games
, 2004
"... The complexity of solving infinite games, including parity, mean payoff, and simple stochastic games, is an important open problem in verification, automata theory, and complexity theory. In this paper we develop an abstract setting for studying and solving such games, as well as related problems, b ..."
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Cited by 6 (0 self)
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The complexity of solving infinite games, including parity, mean payoff, and simple stochastic games, is an important open problem in verification, automata theory, and complexity theory. In this paper we develop an abstract setting for studying and solving such games, as well as related problems, based on function optimization over certain discrete structures. We introduce new classes of completely localglobal (CLG) and recursively localglobal (RLG) functions, and show that strategy evaluation functions for parity and simple stochastic games belong to these classes. We also establish a relation to the previously wellstudied completely unimodal (CU) and localglobal functions. A number of nice properties of CLGfunctions are proved. In this setting, we survey several randomized optimization algorithms appropriate for CU, CLG, and RLGfunctions. We show that the subexponential algorithms for linear programming by Kalai and Matouˇsek, Sharir, and Welzl, can be adapted to optimizing the functions we study, with preserved subexponential expected running time. We examine the relations to two other abstract frameworks for subexponential
A subexponential lower bound for the Random Facet algorithm for Parity Games
"... Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of t ..."
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Cited by 6 (5 self)
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Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of turnbased Stochastic Mean Payoff Games. It is a major open problem whether these game families can be solved in polynomial time. The currently fastest algorithms for the solution of all these games are adaptations of the randomized generalizationof linear programming. We refer to the algorithm ofMatouˇsek, Sharir and Welzl as the Random Facet algorithm. The expected running time of these algorithmsis subexponential in the size of the game, i.e., 2